Dirichlet series
L(s) = 1 | + (0.867 − 0.497i)2-s + (−0.165 + 0.986i)3-s + (0.505 − 0.862i)4-s + (0.209 + 0.977i)5-s + (0.347 + 0.937i)6-s + (−0.999 + 0.0260i)7-s + (0.00975 − 0.999i)8-s + (−0.945 − 0.325i)9-s + (0.668 + 0.744i)10-s + (0.710 + 0.703i)11-s + (0.767 + 0.641i)12-s + (0.964 + 0.263i)13-s + (−0.854 + 0.519i)14-s + (−0.998 + 0.0455i)15-s + (−0.488 − 0.872i)16-s + (−0.0292 − 0.999i)17-s + ⋯ |
L(s) = 1 | + (0.867 − 0.497i)2-s + (−0.165 + 0.986i)3-s + (0.505 − 0.862i)4-s + (0.209 + 0.977i)5-s + (0.347 + 0.937i)6-s + (−0.999 + 0.0260i)7-s + (0.00975 − 0.999i)8-s + (−0.945 − 0.325i)9-s + (0.668 + 0.744i)10-s + (0.710 + 0.703i)11-s + (0.767 + 0.641i)12-s + (0.964 + 0.263i)13-s + (−0.854 + 0.519i)14-s + (−0.998 + 0.0455i)15-s + (−0.488 − 0.872i)16-s + (−0.0292 − 0.999i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(967\) |
Sign: | $0.989 + 0.141i$ |
Analytic conductor: | \(103.918\) |
Root analytic conductor: | \(103.918\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{967} (723, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 967,\ (1:\ ),\ 0.989 + 0.141i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(3.515553966 + 0.2495128674i\) |
\(L(\frac12)\) | \(\approx\) | \(3.515553966 + 0.2495128674i\) |
\(L(1)\) | \(\approx\) | \(1.714580183 + 0.1048913128i\) |
\(L(1)\) | \(\approx\) | \(1.714580183 + 0.1048913128i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.867 - 0.497i)T \) |
3 | \( 1 + (-0.165 + 0.986i)T \) | |
5 | \( 1 + (0.209 + 0.977i)T \) | |
7 | \( 1 + (-0.999 + 0.0260i)T \) | |
11 | \( 1 + (0.710 + 0.703i)T \) | |
13 | \( 1 + (0.964 + 0.263i)T \) | |
17 | \( 1 + (-0.0292 - 0.999i)T \) | |
19 | \( 1 + (0.791 - 0.610i)T \) | |
23 | \( 1 + (0.371 - 0.928i)T \) | |
29 | \( 1 + (0.107 + 0.994i)T \) | |
31 | \( 1 + (0.571 - 0.820i)T \) | |
37 | \( 1 + (0.0422 - 0.999i)T \) | |
41 | \( 1 + (-0.203 - 0.979i)T \) | |
43 | \( 1 + (0.120 + 0.992i)T \) | |
47 | \( 1 + (-0.177 + 0.984i)T \) | |
53 | \( 1 + (0.898 + 0.439i)T \) | |
59 | \( 1 + (-0.628 - 0.777i)T \) | |
61 | \( 1 + (0.746 - 0.665i)T \) | |
67 | \( 1 + (-0.425 + 0.905i)T \) | |
71 | \( 1 + (-0.864 + 0.502i)T \) | |
73 | \( 1 + (0.898 - 0.439i)T \) | |
79 | \( 1 + (-0.216 + 0.976i)T \) | |
83 | \( 1 + (0.719 + 0.694i)T \) | |
89 | \( 1 + (0.996 - 0.0844i)T \) | |
97 | \( 1 + (0.623 + 0.781i)T \) | |
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Imaginary part of the first few zeros on the critical line
−21.67267592313436938689249060896, −20.85459709179740486683314697487, −19.88657764754583240576349217896, −19.41778154096941849509157895960, −18.30464444658969778711113528781, −17.27285330361763235582476142786, −16.7803852773699607786158308821, −16.10259440767054251479795374915, −15.24384289190779111055744754730, −14.00051848833652462934948077983, −13.35045220246645211142562254503, −13.11114103661266793639049999969, −11.98189710643102366429116126254, −11.71247211525057993529533485188, −10.31992064224083832011863313831, −8.9253522258522056884970550089, −8.38169133204447854680231509831, −7.46907287478440887738493486466, −6.24105917735067305875018737398, −6.10850266182740336127126720851, −5.15656167854424395183290197245, −3.795013584743337986194558045989, −3.17669619162966308326882202199, −1.75029238966182566278636989822, −0.83794185066026555825641334142, 0.74109842340404314629584386191, 2.38560705532630587582267928906, 3.12867007443435583409006127754, 3.82452102907586955237016762492, 4.71443044824173678997596337389, 5.78196385745769261951042787516, 6.51196518082494538601247022493, 7.15757603068974111381963669754, 9.118866296171561599774672032118, 9.57776163488735242957147844224, 10.4025071754808066563342432975, 11.13194525198856721057103924991, 11.7689025900195816529963492252, 12.76729291566282564209575396870, 13.83141716426368804537731242636, 14.31206115251739822118021779278, 15.197529236420174013052220511246, 15.85588819817142832152882104106, 16.45924306705683618694466615840, 17.7276705350005306030729837970, 18.58603416796518056783470406323, 19.43805084954337654631421774254, 20.2559934300068676662571954794, 20.83184975990193196108372448688, 21.81836428053861829292305603626